Post on 24-Dec-2015
Two-Process Systems
Companion slides forDistributed Computing
Through Combinatorial TopologyMaurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum
Distributed Computing through Combinatorial Topology
Two-Process Systems
Two-process systems can be captured by elementary
graph theorygentle introduction to more general structures needed
later for larger systemsDistributed Computing through
Combinatorial Topology2
Road MapElementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
Distributed Computing through Combinatorial Topology
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Road MapElementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
Distributed Computing through Combinatorial Topology
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A Vertex
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A VertexCombinatorial: an element of a set.
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A VertexCombinatorial: an element of a set.
Geometric: a point in Euclidean Space
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An Edge
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An EdgeCombinatorial: a set of two vertexes.
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An EdgeCombinatorial: a set of two vertexes.
Geometric: line segment joining two points
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A Graph
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A GraphCombinatorial: a set of sets of vertices.
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A GraphCombinatorial: a set of sets of vertices.
Geometric: points joined by line segments
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Graphs
finite set V with a collection G of subsets of V,
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Graphs
simplices(singular: simplex)
finite set V with a collection G of subsets of V,
vertices
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Graphs
Distributed Computing through Combinatorial Topology
(1) If X 2 G, then |X| · 2
finite set V with a collection G of subsets of V,
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Graphs
Distributed Computing through Combinatorial Topology
(1) If X 2 G, then |X| · 2
finite set V with a collection G of subsets of V,
vertex: |X| = 1 edge: |X|= 2
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Graphs
(1) If X 2 G, then |X| · 2(2) for all v 2 V, {v} 2 G
finite set V with a collection G of subsets of V,
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Graphs
(1) If X 2 G, then |X| · 2(2) for all v 2 V, {v} 2 G
(3) for all X 2 G, and Y ½ X, Y 2 G
finite set V with a collection G of subsets of V,
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Dimension
dim(X) = |X|-1.
dimension 0
dimension 1
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Pure Graphs
pure of dim 0
pure of dim 1
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Graph Coloring
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Graph Coloring
Â: G ! C
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Graph Coloring
Â: G ! Cfor each edge (s0, s1) 2 G, Â(s0) Â(s1).
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Graph Coloring
Â: G ! Cfor each edge (s0, s1) 2 G, Â(s0) Â(s1).
usually process nameschromatic graphs
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Graph Labeling
1 0
0
1
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Graph Labeling
1 0
0
1f: G ! L
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Graph Labeling
1 0
0
1f: G ! Lusually values from some domain
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Labeled Chromatic Graph
0 1
1
0
name(s) = Â(s) view(s) = f(s)
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Simplicial Maps
Vertex-to-vertex map …
that also sends edges to edges.
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Rigid Simplicial Maps
A simplicial map can send an edge to a vertex …
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Rigid Simplicial Maps
A simplicial map can send an edge to a vertex …
A simplicial map that sends distinct vertices to distinct vertices is rigid.
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A Path Between two Vertices
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A Path Between two Vertices
A graph is connected if there is a path between every pair of vertices
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Not Connected
A graph is connected if there is a path between every pair of vertices
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Theorem
Distributed Computing through Combinatorial Topology
Theorem
Á
The image of a connected graph under a simplicial map is connected.
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Carrier Maps
Distributed Computing through Combinatorial Topology
©: G ! 2H
For graphs G, H, a carrier map
Carries each simplex of G to a subgraph of H …
©
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Carrier Maps
Distributed Computing through Combinatorial Topology
©: G ! 2H
For graphs G, H, a carrier map
Carries each simplex of G to a subgraph of H …
©
satisfying monotonicity:for all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).
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Monotonicity
Strict Carrier Maps
Distributed Computing through Combinatorial Topology
For all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).
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Monotonicity
Strict Carrier Maps
Distributed Computing through Combinatorial Topology
For all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).
Equivalent to …©(¾Å¿) µ ©(¾) Å ©(¿)
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Monotonicity
Strict Carrier Maps
Distributed Computing through Combinatorial Topology
For all ¾,¿2G, if ¾µ¿, then ©(¾)µ©(¿).
Equivalent to …©(¾Å¿) µ ©(¾) Å ©(¿)
Definition© is strict if ©(¾Å¿) = ©(¾) Å ©(¿)
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Road MapElementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Two Processes
Distributed Computing through Combinatorial Topology
Hello! I’m
AliceHello! I’m
Bob
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Informal Task Definition
Distributed Computing through Combinatorial Topology
Processes start with input values …
They communicate …
They halt with output values …
legal for those inputs.
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Formal Task Definition
Distributed Computing through Combinatorial Topology
Input graph Iall possible assignments of input values
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Formal Task Definition
Distributed Computing through Combinatorial Topology
Input graph Iall possible assignments of input values
Output graph Oall possible assignments of output values
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Formal Task Definition
Distributed Computing through Combinatorial Topology
Input graph Iall possible assignments of input values
Output graph Oall possible assignments of output values
Carrier map ¢: I ! 2O
all possible assignments of output values for each input
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Task Input Graph: Consensus
Distributed Computing through Combinatorial Topology
1
10
0
I
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Task Input Graph
Distributed Computing through Combinatorial Topology
0 1
10
I
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Task Input Graph
Distributed Computing through Combinatorial Topology
1 1
00
Pure
Colored by process names
Labeled by input values
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Task Output Graph
Distributed Computing through Combinatorial Topology
1 1
00
O
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Task Carrier Map
Distributed Computing through Combinatorial Topology
1 1
00
1 1
00
¢: I ! 2OI O
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Task Carrier Map
Distributed Computing through Combinatorial Topology
1 1
00
1 1
00
¢: I ! 2OI O
If Bob runs alone with input 1 …
then he decides output 1.
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Task Carrier Map
Distributed Computing through Combinatorial Topology
1 1
00
1 1
00
¢: I ! 2OI O
If Bob and Alice both have input 1 …
then they both decide output 1.
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Task Carrier Map
Distributed Computing through Combinatorial Topology
1 1
00
1 1
00
¢: I ! 2OI O
If Bob has 1 and Alice 0 …
then they must agree, on either one.
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Example: Coordinated Attack
Alice and Bob winIf they both attack
together
Alice Bob
Distributed Computing through Combinatorial Topology
Enemy
Indifferent
Attack at dawn!
Attack at noon!
Input Graph
Distributed Computing through Combinatorial Topology
I
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Output Graph
Distributed Computing through Combinatorial Topology
dawn!
noon!
failed!
O
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Carrier Map
Distributed Computing through Combinatorial Topology
¢
I
O
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Carrier Map
Distributed Computing through Combinatorial Topology
¢
dawn!
failed!
dawn! I
O
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Carrier Map
Distributed Computing through Combinatorial Topology
¢
noon!
failed!
noon!I
O
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Carrier Map
Distributed Computing through Combinatorial Topology
¢
dawn!
dawn! I
O
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Example: Coordinated AttackAlice Bob
Distributed Computing through Combinatorial Topology
Enemy
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Example: Coordinated AttackAlice Bob
Distributed Computing through Combinatorial Topology
Enemy
Alice and Bob realize that they do not need to agree
on an exact time …
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Example: Coordinated AttackAlice Bob
Distributed Computing through Combinatorial Topology
Enemy
Alice and Bob realize that they do not need to agree
on an exact time …they will win if attack times
are sufficiently close.
0 1
0 1/5 2/5 3/5 4/5 1
Coordinated Attack Graphs
Distributed Computing through Combinatorial Topology
¢
I
O
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0 1
0 1/5 2/5 3/5 4/5 1
Coordinated Attack Graphs
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¢
I
O
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0 1
0 1/5 2/5 3/5 4/5 1
Coordinated Attack Graphs
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¢
I
O
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0 1
0 1/5 2/5 3/5 4/5 1
Coordinated Attack Graphs
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¢
I
O
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Road MapElementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Protocols
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Models of Computation
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Alice’s Protocol
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shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[A] + mem[B];return δ(view)
Finite program
Bob’s protocol is symmetric
Distributed Computing through Combinatorial Topology
shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[B];return δ(view)
Alice’s Protocol
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shared two-element memory
shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[B];return δ(view)
Alice’s Protocol
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Start with input value
shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[B];return δ(view)
Alice’s Protocol
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Run for L layers
shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[B];return δ(view)
Alice’s Protocol
76Distributed Computing through Combinatorial Topology
Alice writes her value, read Bob’s value, and concatenate it to my view
shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[B];return δ(view)
Alice’s Protocol
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Alice writes her value, read Bob’s value, and concatenate it to my view
(full-information protocol)
shared mem array 0..1 of Valueview: Value := my input value;for i: int := 0 to L do mem[A] := view; view := view + mem[B];return δ(view)
Alice’s Protocol
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finally, apply task-specific decision map to view
Formal Protocol Definition
Distributed Computing through Combinatorial Topology
Input graph Iall possible assignments of input values
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Formal Protocol Definition
Distributed Computing through Combinatorial Topology
Input graph Iall possible assignments of input values
Protocol graph Pall possible process views after execution
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Formal Protocol Definition
Distributed Computing through Combinatorial Topology
Input graph Iall possible assignments of input values
Protocol graph Pall possible process views after execution
Carrier map ¥: I ! 2P
all possible assignments of views
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One-Round Protocol Graph
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10
0? 0101 ?1
¥
I
P
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One-Round Protocol Graph
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0? 0101 ?1
Colored by process names
Labeled with final views
P
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One-Round Protocol Graph
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0? 0101 ?1
Alice finishes before Bob starts, doesn’t see his value
P
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One-Round Protocol Graph
Distributed Computing through Combinatorial Topology
0? 0101 ?1
Alice and Bob run together,she sees his value.
P
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One-Round Protocol Graph
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0? 0101 ?1
Alice finishes, then Bob starts
P
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One-Round Protocol Graph
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0? 0101 ?1
Alice and Bob run together
P
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One-Round Protocol Graph
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0? 0101 ?1
Bob can’t tell whether Alice saw him
P
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Execution Carrier Map
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0? 0101 ?1
¥
I
P
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Execution Carrier Map
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0? 0101 ?1
¥
I
P
¥: I ! 2P
strict carrier map¥(¾) Å ¥(¿) = ¥(¾ Å ¿)90
Output graph0 2/31/3 1
0? 0101 ?1 Protocol graph
δ
The Decision Map
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δ
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All Together
Distributed Computing through Combinatorial Topology
0 1
0 2/31/3 1
0? 0101 ?1¢
δ
I
P
O
¥
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Definition
Distributed Computing through Combinatorial Topology
Decision map δ is carried by carrier map ¢ if
for each input vertex s,
for each input edge ¾,
δ(¥(s)) µ ¢(s)
δ(¥(¾)) µ ¢(¾).
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Meaning
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δ(¥(s)) µ ¢(s)
process strarts in state s
Meaning
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δ(¥(s)) µ ¢(s)
runs the protocol to completion
Meaning
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δ(¥(s)) µ ¢(s)
makes a decision …
Meaning
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δ(¥(s)) µ ¢(s)
decision is permitted by task carrier map
Definition
Solving a Task
Distributed Computing through Combinatorial Topology
The protocol (I,P,¥) solves the task (I, O, ¢)
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Definition
Solving a Task
Distributed Computing through Combinatorial Topology
The protocol (I,P,¥) solves the task (I, O, ¢)
if there is …
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Definition
Solving a Task
Distributed Computing through Combinatorial Topology
The protocol (I,P,¥) solves the task (I, O, ¢)
if there is …
a simplicial decision mapδ:P ! O
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Definition
Solving a Task
Distributed Computing through Combinatorial Topology
The protocol (I,P,¥) solves the task (I, O, ¢)
if there is …
a simplicial decision mapδ:P ! O
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such that δ is carried by ¢.(δ agrees with ¢)
Layered Read-Write Model
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Layered Read-Write Protocol
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shared mem array 0..1,0..L of Valueview: Value := my input value;for i: int := 0 to L do mem[i][A] := view; view := view + mem[A] + mem[B];return δ(view)
shared mem array 0..1,0..L of Valueview: Value := my input value;for i: int := 0 to L do mem[i][A] := view; view := view + mem[A] + mem[B];return δ(view)
As before, run for L layers
Layered Read-Write Protocol
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shared mem array 0..1,0..L of Valueview: Value := my input value;for i: int := 0 to L do mem[i][A] := view; view := view + mem[A] + mem[B];return δ(view)
Layered Read-Write Protocol
Distributed Computing through Combinatorial Topology
Each layer uses a distinct, “clean” memory
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Layered R-W Protocol Graph
Distributed Computing through Combinatorial Topology
10
0? 0101 ?1
¥
I
P
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Layered R-W Protocol Graph
Distributed Computing through Combinatorial Topology
¥
P is always a subdivision of I
I P
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Road MapElementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
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Alice’s 1/3-Agreement Protocol
Distributed Computing through Combinatorial Topology
mem[A] := 0other := mem[B]if other == ? then decide 0else decide 1/3
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Alice’s 1/3-Agreement Protocol
Distributed Computing through Combinatorial Topology
mem[A] := 0if mem[B] == ? then decide 0else decide 1/3
Alice writes her value to memory
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Alice’s 1/3-Agreement Protocol
Distributed Computing through Combinatorial Topology
mem[A] := 0if mem[B] == ? then decide 0else decide 1/3
If she doesn’t see Bob’s value, decide her own.
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Alice’s 1/3-Agreement Protocol
Distributed Computing through Combinatorial Topology
mem[A] := 0if mem[B] == ? then decide 0else decide 1/3
If she see’s Bob’s value, jump to the middle
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0 2/31/3 1
0 2/31/3 1
0 2/31/3 1
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One-Layer 1/3-Agreement Protocol
Distributed Computing through Combinatorial Topology
0 1
0 2/31/3 1
0? 0101 ?1
δ
I
P
O
¥
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No 1-Layer 1/5-Agreement Protocol
Distributed Computing through Combinatorial Topology
0 1
1/5 3/52/5 4/5
0? 0101 ?1
δ
P
O
¥
10
(no map possible)
I
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10
0 1/5 2/5 3/5 4/5 1
¥
0? 0101 ?1
2-Layer 1/5-Agreement
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¥
I
O
layer 1
layer 2
δ
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Fact
Distributed Computing through Combinatorial Topology
In the layered read-write model,
The 1/K-Agreement Task
Has a dlog3 Ke–layer protocol
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Road MapElementary Graph Theory
Tasks
Models of Computation
Approximate Agreement
Task Solvability
Distributed Computing through Combinatorial Topology
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Fact
Distributed Computing through Combinatorial Topology
The protocol graph for any L-layer protocol with input graph I is a subdivision of I, where each edge is subdivided 3L times.
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Main Theorem
Distributed Computing through Combinatorial Topology
The two-process task (I, O, ¢) is solvable in the layered read-write model if and only if there exists a connected carrier map ©: I ! 2O carried by ¢.
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Corollary
Distributed Computing through Combinatorial Topology
The consensus task has no layered read-write protocol
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Corollary
Distributed Computing through Combinatorial Topology
Any ²–agreement task has a layered read-write protocol
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Distributed Computing through Combinatorial Topology